I'm not sure I understand some of the above answers.
Firstly, the last pay date is not relevant - it is the date of the contract termination of the end of August that is key.
Secondly, you have a continuing contract until the end of August - the way the employer requires you to operate means that you do not require to attend at school after 13th July. So in theory you could do other work from then.
Thirdly, look in your terms to check that it is not a condition that you do not have another paid job without the permission of your employer - it is in some people's contracts. It is this term that might catch you out. If you cannot find such a term, I would be inclined to write to the employer as follows: -
Dear Mr Employer, as you know I am employed with you until 31st August when you are making me redundant. I cannot find any term that prevents me from taking a second job whilst in your employment and I propose to start a second job from xyz of July. Please confirm that this is acceptable to you. [or words to that effect - you get the drift]
Fourthly, I haven't a clue what Dot's point about NI is all about.